Flattone
Flattone is an alternative extension to 5-limit meantone, the temperament that tempers out the syntonic comma (81/80). It is generated by a fifth that is typically flatter than that of septimal meantone, and nine of those reach the pitch class of 8/7, so that 7/4 is a diminished seventh (C–B𝄫), 7/6 is a diminished third (C–E𝄫), and 7/5 is a doubly diminished fifth (C–G𝄫). Although 7/4 is simpler than in septimal meantone, the full 9-odd-limit tonality diamond is more complex as the 5 and 7 are reached by going in opposite directions, while also being less accurate.
However, it makes up for that by having simpler 11- and 13-limit interpretations – the whole tone is now flat enough that it can function as 9/8, 10/9 and 11/10, tempering out 100/99 and making 11/8 an augmented fourth (C–F#). This means the major third functions as both 5/4 and 11/9. Tempering out 65/64 means it also represents their mediant 16/13, making 13/8 a minor sixth (C–A♭) and a full otonal chord of 8:9:10:11:12:13:14:15:16 accessible with a gamut of 16 notes, compared to 19 for tridecimal meantone or the 29 required by meanpop.
Reasonable tunings lie between 19edo and 26edo. 19edo is the point where 7/4 and 12/7 are conflated. Any tuning whose fifth is sharper than 19edo's has the sizes of 7/4 and 12/7 inverted, so they can be more properly analysed as septimal meantone. Similarly, 26edo is the point where 7/5 and 10/7 are conflated. Any tuning whose fifth is flatter than 26edo's has the sizes of 7/5 and 10/7 inverted, so they can be more properly analysed as a flatter-than-flattone temperament.
See Meantone family #Flattone for technical data.
Interval chain
In the following table, odd harmonics 1–13 are in bold.
| # | Cents* | Approximate ratios |
|---|---|---|
| 0 | 0.0 | 1/1 |
| 1 | 693.0 | 3/2 |
| 2 | 186.1 | 9/8, 10/9, 11/10 |
| 3 | 879.1 | 5/3 |
| 4 | 372.1 | 5/4, 16/13, 26/21 |
| 5 | 1065.1 | 11/6, 13/7, 15/8, 24/13 |
| 6 | 558.2 | 11/8, 18/13 |
| 7 | 51.2 | 25/24, 27/26, 33/32, 36/35, 55/54, 64/63 |
| 8 | 744.2 | 20/13, 32/21 |
| 9 | 237.3 | 8/7, 15/13 |
| 10 | 930.3 | 12/7, 22/13 |
| 11 | 423.3 | 9/7 |
| 12 | 1116.4 | 27/14, 40/21 |
| 13 | 609.4 | 10/7 |
* In 13-limit CTE tuning, octave reduced
As a detemperament of 7et
Flattone is best analyzed as a 7-form system. It is melodically intuitive compared to standard meantone, in that 7-limit intervals are found as augmented and diminished versions of the category they "should" belong to (that is, the quartertone represents both 25/24 and 64/63).
| Interval category | -2 quartertones | -1 quartertone | 0 quartertones | 1 quartertone | 2 quartertones | |||||
|---|---|---|---|---|---|---|---|---|---|---|
| Cents* | Approx. ratios | Cents* | Approx. ratios | Cents* | Approx. ratios | Cents* | Approx. ratios | Cents* | Approx. ratios | |
| Unison | 1098 | 28/15 | 1149 | 48/25, 52/27, 64/33, 35/18, 98/55, 63/32 | 0 | 1/1 | 51 | 25/24, 27/26, 33/32, 36/35, 55/54, 64/63 | 102 | 15/14 |
| Second | 84 | 28/27, 21/20 | 135 | 16/15, 12/11, 13/12, 14/13 | 186 | 9/8, 10/9, 11/10 | 237 | 8/7, 15/13 | 288 | |
| Third | 219 | 270 | 7/6 | 321 | 6/5 | 372 | 5/4, 16/13, 26/21 | 423 | 9/7 | |
| Fourth | 405 | 456 | 13/10, 21/16 | 507 | 4/3 | 558 | 11/8, 18/13 | 609 | 10/7 | |
| Fifth | 591 | 7/5 | 642 | 16/11, 13/9 | 693 | 3/2 | 744 | 20/13, 32/21 | 795 | |
| Sixth | 777 | 14/9 | 828 | 8/5, 13/8, 21/13 | 879 | 5/3 | 930 | 12/7 | 981 | |
| Seventh | 912 | 963 | 7/4, 26/15 | 1014 | 9/5, 16/9 | 1065 | 15/8, 11/6, 13/7, 24/13 | 1116 | 40/21, 27/14 | |
Scales
- Flattone12 – 12-tone chromatic scale in 13-limit POTE tuning
Tunings
Tuning spectrum
| Edo generator |
Eigenmonzo (unchanged-interval)* |
Generator (¢) |
Comments |
|---|---|---|---|
| 64/63 | 689.609 | ||
| 13/8 | 689.868 | ||
| 11/6 | 689.873 | ||
| 19\33 | 690.909 | ||
| 13/11 | 691.079 | ||
| 21/16 | 691.152 | ||
| 9/5 | 691.202 | 1/2 comma | |
| 53\92 | 691.304 | ||
| 21/11 | 691.467 | ||
| 34\59 | 691.525 | ||
| 49\85 | 691.765 | ||
| 11/8 | 691.886 | ||
| 11/7 | 692.166 | 11- and 13-odd-limit minimax | |
| 13/12 | 692.285 | ||
| 15\26 | 692.308 | Lower bound of 7-, 9-, 11-, and 13-odd-limit diamond monotone | |
| 7/4 | 692.353 | ||
| 21/13 | 692.437 | ||
| 36/35 | 692.681 | ||
| 49/48 | 692.858 | ||
| 41\71 | 692.958 | ||
| 21/20 | 692.961 | ||
| 13/10 | 693.223 | ||
| 7/6 | 693.313 | ||
| 26\45 | 693.333 | ||
| 7/5 | 693.653 | 7-odd-limit minimax | |
| 37\64 | 693.750 | ||
| 9/7 | 694.099 | 9-odd-limit minimax | |
| 15/13 | 694.193 | ||
| 15/14 | 694.246 | ||
| 13/7 | 694.340 | ||
| 11\19 | 694.737 | Upper bound of 7-, 9-, 11-, 13-odd-limit diamond monotone | |
| 5/3 | 694.786 | 1/3 comma | |
| 25/24 | 695.810 | 2/7 comma | |
| 5/4 | 696.578 | 1/4 comma, 5-odd-limit minimax | |
| 15/8 | 697.654 | 1/5 comma | |
| 7\12 | 700.000 | ||
| 3/2 | 701.955 | Pythagorean tuning |
* Besides the octave